most mansion/apt don't allow pets, as what Nyororin explained. But Lots of tenants do have pets. And they are Japanese Tenants not foreigners.
The thing is, as long as you don't report them they will not report you. But first you have to observe your neighbors. And I agree with everybody else, don't bring your dog, its both tiring for you and your beloved pet. Though you can maybe apply as a volunteer to shelter dogs that don't have a home yet or no longer have a home.